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I have read several places that Descartes' Rule of Signs was familiar to both Descartes and Newton, and that both considered it too "obvious" to merit a proof. I know how to prove it, but I would like to know how they intuitively sensed that it was true. Newton apparently used it in one of his books, so he must have had a good reason to believe it was true if he never bothered to attempt a proof.

Just for clarification, I am referring to the theorem that the number of positive roots of the polynomial $$p(x)=a_nx^n+⋯+a_1x+a_0$$ is equal to or less than by an even number the number of sign changes in p as written in the order above (descending powers of x).

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Basically, at different values of $x$ different terms in the polynomial "dominate." So every the sign switches, there will be a change in the direction of the curve. Either (1) this will result in crossing the x-axis and a root or (2) there will have to be another change, meaning "losing roots" will always happen in pairs. So the roots are equal to or less than by something x2 the number of sign changes.

Also, have you tried looking at the wikipedia page?

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You might be interested in Michael Bensimhoun's article:

Historical account and ultra-simple proofs of Descartes's rule of signs

I've just had a quick look at it and it seems very accessible.

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    $\begingroup$ Thanks for the link. It seems that they assume Descartes to have used some sort of inductive argument, and then they present a 'simpler' version, but their proof is quite tedious (as would any inductive be) and I'm having doubts that this could have been the intuition that they knew of. $\endgroup$ – user141733 Apr 8 '14 at 19:15
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The intuition is that each $x^k$ with a different sign than the previous summands may outweigh the higher powers for small $x$, but not for large $x$. Of course, it is imaginable that the "struggle" between these two is more complicated - but it is not. A rigorous proof would of course be preferable.

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